Vishal Patel^{1} and Noriko Salamon^{1}

We demonstrate a novel technique for studying white matter pathology by examining the statistical properties of the DWI signal. We apply a sparse coding method, K-SVD, to decompose a diffusion-weighted series. We then quantify the efficiency of the resulting encoding by computing the Gini coefficient. We show that this measure is abnormally decreased in a cohort of lissencephaly patients compared to age-matched control subjects. Our results support the hypotheses that more organized white matter can be more sparsely encoded and that the sparsity of the encoding may thus be used to infer pathological white matter states.

We queried our institutional database for all brain MR studies performed over the 60-month period ending in October 2016 that included multidirectional diffusion-weighted imaging. Only studies performed on individuals younger than 21 years of age and reported as either normal or lissencephaly were evaluated. For each lissencephaly case, the two closest age-matched controls were included for further comparison. Diffusion-weighted images were acquired at 1.5 T with *b* = 1200 s/mm^{2} over 30 gradient directions. Images were corrected for head motion and eddy current effects, and fractional anisotropy maps were computed^{1} and thresholded (FA > 0.2) to produce a white matter mask.

Each diffusion-weighted series was decomposed using K-SVD^{2}, a coding method which simultaneously seeks both an optimal dictionary $$$\mathbf{D}$$$ and sparse coefficients $$$\mathbf{X}$$$ that approximate the observed data $$$\mathbf{Y}$$$ subject to a sparsity constraint $$$T_0$$$ on each coefficient vector $$$\mathbf{x}_p$$$:

$$\DeclareMathOperator{\argmin}{argmin} \underset{\mathbf{D},\mathbf{X}}{\argmin} \lVert\mathbf{Y} - \mathbf{DX}\rVert^2_\mathrm{F} \quad \text{s.t.} \quad \forall p,\, \lVert\mathbf{x}_p\rVert_0 \leq T_0$$

We summarize the K-SVD algorithm, initially proposed for DWI denoising, in Figure 1 and refer the reader to previous descriptions^{3,4} for implementation details. As in prior works utilizing K-SVD for DWI encoding, a fixed dictionary size ($$$K=200$$$) and sparsity threshold ($$$T_0=10$$$) were used throughout this study.

The sparsity of the encoding was quantified by computing the Gini coefficient^{5}, previously validated in this context^{6}, for each coding vector:

$$\textrm{Gini Coefficient}\left(\mathbf{x}\right)=1-2\sum^{K}_{k=1}\frac{\mathbf{x}\left[k\right]}{\lVert \mathbf{x} \rVert_1} \left( \frac{K-k+\frac{1}{2}}{K} \right)$$

The Gini coefficient takes values over the interval $$$\left[0, 1\right]$$$, with unity representing a maximally sparse coding vector containing a single nonzero element. This sparsity measure has several important advantages^{7} over the $$$\ell^0$$$ and $$$\ell^1$$$ norms, including invariance to constant offset and scaling, and has previously been shown to increase monotonically thoughout normal pediatric white matter maturation^{6}.

For each subject, the mean Gini coefficient was calculated across the entire segmented white matter, and the distribution of these means was compared between the lissencephaly and control sample populations using a two-sample *t*-test.

We have shown for the first time that the sparsity of the diffusion-weighted MR signal, quantified as the Gini coefficient derived from a K-SVD encoding, can be used directly to identify pathological white matter conditions such as lissencephaly. This technique provides a means for quickly identifying potential disease states without requiring the manual investigation of particular brain regions and avoids the biases and assumptions inherent in diffusion model fitting and tractography.

1. Basser PJ, Pierpaoli C. Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. Journal of Magnetic Resonance B. 1996;111(3):209–19.

2. Elad M, Aharon M. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Transactions on Image Processing. 2006;15(12):3736–45.

3. Rubinstein R, Zibulevsky M, Elad M. Effcient implementation of the K-SVD algorithm using batch orthogonal matching pursuit. Technical Report. CS Technion. 2008.

4. Patel V, Shi Y, Thompson PM, Toga AW. K-SVD for HARDI denoising. IEEE International Symposium on Biomedical Imaging. 2011:1805–08.

5. Dixon PM, Weiner J, Mitchell-Olds T, Woodley R. Bootstrapping the Gini coeffcient of inequality. Ecology. 1987;68:1548–51.

6. Patel V, Fitzgibbons M, Thompson PM, Toga AW, Salamon N. Examining global white matter development via the sparse coding properties of diffusion-weighted MRI. International Society for Magnetic Resonance in Medicine. 2016.

7. Hurley N, Rickard S. Comparing measures of sparsity. IEEE Transactions on Information Theory. 2009;55(10):4723–41.

Figure 1: K-SVD for diffusion-weighted MRI denoising. The K-SVD algorithm illustrated as pseudocode (left) and schematically (right). Lowercase symbol subscripts and superscripts denote the column and row vectors, respectively, extracted from the corresponding matrices (uppercase symbols). We refer to dedicated prior reports^{3,4} for a comprehensive description and implementation details.

Figure 2: Representative axial directionally-encoded color images from an 18-month old lissencephaly patient (left) and age-matched control subject (right) demonstrating the severe white matter derangement. Voxels colored red, green, and blue contain fibers oriented principally in the medial-lateral, anterior-posterior, and superior-inferior directions, respectively. Color intensity is scaled by the fractional anisotropy.

Figure 3: Summary of subject demographic characteristics. The study population was comprised of 7 lissencephaly patients and 14 age-matched controls. Neither age nor gender distribution differed significantly between the groups.